Springer Series in Solid and StructuralMechanics

Volume 13

Series Editors

Michel Frémond, Rome, Italy

Franco Maceri, Department of Civil Engineering and Computer Science,University of Rome “Tor Vergata”, Rome, Italy

The Springer Series in Solid and Structural Mechanics (SSSSM) publishes newdevelopments and advances dealing with any aspect of mechanics of materials andstructures, with a high quality. It features original works dealing with mechanical,mathematical, numerical and experimental analysis of structures and structuralmaterials, both taken in the broadest sense. The series covers multi-scale, multi-fieldand multiple-media problems, including static and dynamic interaction. It alsoillustrates advanced and innovative applications to structural problems from scienceand engineering, including aerospace, civil, materials, mechanical engineering andliving materials and structures. Within the scope of the series are monographs,lectures notes, references, textbooks and selected contributions from specializedconferences and workshops.

More information about this series at http://www.springer.com/series/10616

Friedel Hartmann • Peter Jahn

Statics and InfluenceFunctionsFrom a Modern Perspective

Second Edition

123

Friedel HartmannInstitute of Structural MechanicsUniversity of KasselKassel, Germany

Peter JahnInstitute of Structural MechanicsUniversity of KasselKassel, Germany

ISSN 2195-3511 ISSN 2195-352X (electronic)Springer Series in Solid and Structural MechanicsISBN 978-3-030-55888-8 ISBN 978-3-030-55889-5 (eBook)https://doi.org/10.1007/978-3-030-55889-5

1st edition: © Springer International Publishing AG 20172nd edition: © The Editor(s) (if applicable) and The Author(s), under exclusive license to SpringerNature Switzerland AG 2021This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whetherthe whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse ofillustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, andtransmission or information storage and retrieval, electronic adaptation, computer software, or by similaror dissimilar methodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, expressed or implied, with respect to the material containedherein or for any errors or omissions that may have been made. The publisher remains neutral with regardto jurisdictional claims in published maps and institutional affiliations.

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Preface to the Second Edition

This second edition of Statics and Influence Functions—from a Modern Perspectivehas been completely revised and new material has been added to extend the scopeof the book.

The list of the differential equations in the first chapter has been more thandoubled and the second chapter on Betti’s theorem includes new sections on adjointoperators and St. Venant’s principle.

The equivalent stress transformation in chapter three is a novel approach tocouple diverse structural elements.

In the fourth chapter, sections on the patch test, pollution and super convergencewere added.

In the fifth chapter on reanalysis, new sections on adjoint sensitivity analysis viaduality techniques investigate the close connection between reanalysis with Green’sfunctions, optimization and parameter identification.

A new direct approach to calculate the modified displacement vector withoutsolving the full system is introduced in chapter five. Practical application of thistechnique is one-click-reanalysis, which allows modifications of a structure to bemade with single mouse clicks.

The mixed and nonlinear problems each have been given separate chapters,seven and eight, to allow for more specifics.

In the ninth chapter, a section onSobolev’s EmbeddingTheoremwas added to allowa more detailed exposition of the practical implications of this important theorem.

The new edition offers four programs for download, see Chap. 10: Solution of thePoisson equation, 2-D elasticity, Kirchhoff plates and planar frames. These are generalpurpose programs, but with a particular emphasis on influence functions to put thetheory into practice. The frame program also demonstrates one-click-reanalysis.

Kassel, Germany Friedel HartmannJune 2020 Peter Jahn

v

Preface to the First Edition

The new is the old and the old is mightier than ever before.

The subject of this book are influence functions and the role they play in finiteelement analysis of structures. Influence functions are a classical tool of structuralanalysis and are dearly loved by “old-school” engineers since with some cleversketches—if need be on a beer mat—it is easy to understand the behavior of astructure or to find the weak spots in a design.

Unfortunately, with the advent of finite element programs though, the applicationof influence functions has faded into the background. When in doubt, one ratherstudies variants in a design with the computer than strive for a deeper understandingwhich the study of influence functions can provide so easily and so well.

But new results have rekindled the interest in influence functions because weknow today that in linear analysis, finite elements compute “everything”—as we aretempted to say—with influence functions. This equals a loop backward. Classicalhand methods seemed outdated and old fashioned but in finite elements, they haverisen like Phoenix from the ashes. FE-analysis is more classical than we everimagined.

In the olden days, the subject of influence functions mostly focused on theanalysis of frames with the Müller-Breslau principle but in FE-analysis, the conceptof influence functions has a much broader and wider scope.

The key word is functionals.The deflection at the midspan of a beam, the bending moment at a fixed edge, the

force in a pier, all these are functionals. Anything you can calculate with finiteelements is considered a functional. And to each linear functional belongs a Green’sfunction, an influence function.

Influence functions, too, are displacements; they are the reaction of a structure tospecial point loads, to Dirac deltas, but normally a mesh is not that flexible enoughto generate the exact intricate shape and exact peaks of these influence functions.This is the reason why FE-results are not exact. FE-programs operate withapproximate, substitute influence functions.

The influence functions are the “real” shape functions, the physical shapefunctions. A good approximation of these functions is the key to good FE-results.

So, in structural analysis—and we dare say all of linear computationalmechanics—influence functions play a dominant role. This is why we have writtenthis book.

vii

It is not a book for a first course in structural mechanics; the reader should beacquainted with the basic principles and the reader should have seen influencefunctions in action.

We treat the subject also, as it seems, with a rather sharp pencil but this is moreor less self-defense because in the advent of time, many things in structural analysishave so much settled in that it is hard to discover the mathematics behind theformulas—all too often the ubiquitous dWe ¼ dWi is considered sufficient proof.

Kassel, Germany Friedel HartmannOctober 2016 Peter Jahn

PS. The original title of the book was Influence Functions—from a ModernPerspective but in the time of search engines, it seemed better to extend the title toStatics and Influence Functions—from a Modern Perspective. In doing so, it wasnot our intention to construct a (non-existing) contrast between old and new statics.Only the view on influence functions has changed with the advent of the computer.

viii Preface to the First Edition

Contents

1 Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Principle of Virtual Displacements . . . . . . . . . . . . . . . 21.1.3 Betti’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.4 Influence Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.5 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Green’s Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.1 Longitudinal Displacement uðxÞ of a Bar . . . . . . . . . . . 91.2.2 Shear Deformation wSðxÞ of a Beam . . . . . . . . . . . . . . 111.2.3 Deflection w of a Rope . . . . . . . . . . . . . . . . . . . . . . . . 111.2.4 Deflection w of a Beam . . . . . . . . . . . . . . . . . . . . . . . 121.2.5 Deflection w of a Beam, 2nd Order Theory . . . . . . . . . 121.2.6 Beam on an Elastic foundation . . . . . . . . . . . . . . . . . . 131.2.7 Tensile Chord Bridge . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.8 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Variational Principles of Structural Analysis . . . . . . . . . . . . . . . 141.4 Zero Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.5.1 The Principle of Virtual Displacements . . . . . . . . . . . . 181.5.2 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . 211.5.3 The Principle of Virtual Forces . . . . . . . . . . . . . . . . . . 21

1.6 Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.7 Spring Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.8 Single Forces and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.9 Support Settlements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.10 Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.11 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.12 The Complete Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

ix

1.13 Shortcuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.14 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.15 Mohr Versus Betti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.16 Weak and Strong Influence Functions . . . . . . . . . . . . . . . . . . . 351.17 The Canonical Boundary Values . . . . . . . . . . . . . . . . . . . . . . . 391.18 The Reduction of the Dimension . . . . . . . . . . . . . . . . . . . . . . . 411.19 Boundary Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.20 Finite Elements and Boundary Elements . . . . . . . . . . . . . . . . . 461.21 Test Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481.22 Do Virtual Displacements Have to Be Small? . . . . . . . . . . . . . . 491.23 Only When in Equilibrium? . . . . . . . . . . . . . . . . . . . . . . . . . . . 511.24 What Counts as Displacement and What as Force? . . . . . . . . . . 521.25 The Number of Force and Displacement Terms . . . . . . . . . . . . 521.26 Why the Minus in �H w} ¼ p? . . . . . . . . . . . . . . . . . . . . . . . . 531.27 The Virtual Internal Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 531.28 Castigliano’s Theorem(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541.29 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561.30 The Mathematics Behind the Equilibrium Conditions . . . . . . . . 591.31 Balance and Second-Order Theory . . . . . . . . . . . . . . . . . . . . . 591.32 Sources and Sinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601.33 The Principle of Minimum Potential Energy . . . . . . . . . . . . . . . 61

1.33.1 Minimum or Maximum? . . . . . . . . . . . . . . . . . . . . . . . 621.33.2 Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651.33.3 The Size of the Trial Space V . . . . . . . . . . . . . . . . . . . 66

1.34 Infinite Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681.35 Sobolev’s Embedding Theorem . . . . . . . . . . . . . . . . . . . . . . . . 711.36 Reduction Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741.37 The Force Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761.38 Where Does It Run To? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771.39 Finite Elements and Green’s Identity . . . . . . . . . . . . . . . . . . . . 79References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2 Betti’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812.2 Influence Functions for Displacements . . . . . . . . . . . . . . . . . . . 83

2.2.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852.3 Influence Functions for Forces . . . . . . . . . . . . . . . . . . . . . . . . . 88

2.3.1 Influence Function for NðxÞ . . . . . . . . . . . . . . . . . . . . 902.3.2 Influence Function for MðxÞ . . . . . . . . . . . . . . . . . . . . 912.3.3 Influence Function for VðxÞ . . . . . . . . . . . . . . . . . . . . 912.3.4 Settlement of a Support . . . . . . . . . . . . . . . . . . . . . . . 922.3.5 Temperature Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 942.3.6 Single Moments Differentiate the Influence

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

x Contents

2.4 Statically Determinate Structures . . . . . . . . . . . . . . . . . . . . . . . 942.4.1 Pole-Plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 962.4.2 Construction of Pole-Plans . . . . . . . . . . . . . . . . . . . . . 972.4.3 How to Determine the Magnitude of Rotations . . . . . . 982.4.4 Influence Function for a Shear Force, Fig. 2.17 . . . . . . 1002.4.5 Influence Function for a Normal Force, Fig. 2.18 . . . . . 1002.4.6 Influence Function for a Moment, Fig. 2.19 . . . . . . . . . 1022.4.7 Influence Function for a Moment, Fig. 2.20 . . . . . . . . . 1032.4.8 Influence Function for a Shear Force, Fig. 2.21 . . . . . . 1042.4.9 Influence Function for Two Support Reactions,

Fig. 2.22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1042.4.10 Abutment Reaction, Fig. 2.23 . . . . . . . . . . . . . . . . . . . 104

2.5 Free Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082.6 Statically Indeterminate Structures . . . . . . . . . . . . . . . . . . . . . . 1092.7 Influence Functions for Support Reactions . . . . . . . . . . . . . . . . 1112.8 Jumps in Internal Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1122.9 The Zeros of the Shear Force . . . . . . . . . . . . . . . . . . . . . . . . . 1142.10 Dirac Deltas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1172.11 Dirac Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1192.12 Point Values in 2-D and 3-D . . . . . . . . . . . . . . . . . . . . . . . . . . 1252.13 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1262.14 The Adjoint Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1282.15 Monopoles and Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1292.16 The Leaning Tower of Pisa . . . . . . . . . . . . . . . . . . . . . . . . . . . 1342.17 Influence Functions for Integral Values . . . . . . . . . . . . . . . . . . 1372.18 Influence Functions Integrate . . . . . . . . . . . . . . . . . . . . . . . . . . 1422.19 St. Venant’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1442.20 Second-Order Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

3 Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493.1 The Minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1503.2 Why the Nodal Values of the Rope Are Exact . . . . . . . . . . . . . 1523.3 Adding the Local Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553.4 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1573.5 Equivalent Nodal Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1593.6 Fixed End Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1603.7 Shape Forces and the FE-load . . . . . . . . . . . . . . . . . . . . . . . . . 1623.8 How the Ball Got Rolling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1673.9 Assembling the Element Matrices . . . . . . . . . . . . . . . . . . . . . . 1683.10 Equivalent Stress Transformation . . . . . . . . . . . . . . . . . . . . . . . 1693.11 Calculation of Influence Functions with Finite Elements . . . . . . 1753.12 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1783.13 Generalized Influence Functions . . . . . . . . . . . . . . . . . . . . . . . 180

Contents xi

3.14 Weak and Strong Influence Functions . . . . . . . . . . . . . . . . . . . 1803.15 The Local Influence Function . . . . . . . . . . . . . . . . . . . . . . . . . 1893.16 The Central Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1923.17 Representation of an FE-solution . . . . . . . . . . . . . . . . . . . . . . . 1973.18 Frame Structures and J ¼ gT f . . . . . . . . . . . . . . . . . . . . . . . . . 1983.19 State Vectors and Measurements . . . . . . . . . . . . . . . . . . . . . . . 1983.20 Maxwell’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2003.21 The Inverse Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 2033.22 The Trial Space Vh Has Two Bases . . . . . . . . . . . . . . . . . . . . . 2093.23 General Form of an FE-Influence Function . . . . . . . . . . . . . . . . 2093.24 The Dominance of the Columns gi of the Inverse . . . . . . . . . . . 2113.25 Nature Makes No Jumps, but Finite Elements Do . . . . . . . . . . . 2123.26 The Path from the Source Point to the Load . . . . . . . . . . . . . . . 2133.27 The Columns of K and K�1 . . . . . . . . . . . . . . . . . . . . . . . . . . 2153.28 The Inverse as an Analysis Tool . . . . . . . . . . . . . . . . . . . . . . . 2193.29 Local Changes and the Inverse . . . . . . . . . . . . . . . . . . . . . . . . 2203.30 Mohr and the Flexibility Matrix K�1 . . . . . . . . . . . . . . . . . . . . 2203.31 Non-uniform Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2213.32 Sensitivity Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2223.33 Support Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2243.34 If a Support Settles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2303.35 Influence Function for a Rigid Support . . . . . . . . . . . . . . . . . . 2323.36 Shear Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2353.37 Influence Function for an Elastic Support . . . . . . . . . . . . . . . . . 2363.38 Elasticity Theory and Point Supports . . . . . . . . . . . . . . . . . . . . 2373.39 Point Supports Are Hot Spots . . . . . . . . . . . . . . . . . . . . . . . . . 2393.40 The Amputated Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2423.41 A Dipole at the Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2453.42 Single Force at a Node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2453.43 Predeformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2503.44 The Limits of FE-Influence Functions . . . . . . . . . . . . . . . . . . . 2523.45 Checking Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2533.46 Transient Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2553.47 The Intelligence of Functions . . . . . . . . . . . . . . . . . . . . . . . . . 255References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

4 Betti Extended . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2594.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2604.2 At Which Points Is the FE-Solution Exact? . . . . . . . . . . . . . . . 2624.3 Exact Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2674.4 One-Dimensional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 2674.5 Isogeometric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2694.6 Planar Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

xii Contents

4.7 Point Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2734.8 If the Solution Lies in Vh . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2744.9 Patch Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2764.10 Adaptive Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2774.11 Pollution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

4.11.1 Causes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2834.11.2 Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

4.12 Super Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

5 Stiffness Changes and Reanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . 2915.1 Parameter Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2935.2 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2935.3 Adding or Subtracting Stiffness . . . . . . . . . . . . . . . . . . . . . . . . 2965.4 Dipoles and Monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2975.5 Displacements and Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2995.6 Symmetry and Antisymmetry . . . . . . . . . . . . . . . . . . . . . . . . . 3015.7 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3015.8 The Effects Fade Away . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3035.9 The Relevance of These Results . . . . . . . . . . . . . . . . . . . . . . . 3045.10 Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3075.11 Forces jþ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3085.12 Replacement as Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . 3115.13 The Derivative of the Inverse K�1 . . . . . . . . . . . . . . . . . . . . . . 3125.14 The Derivatives @u=@fk and @u=@kij . . . . . . . . . . . . . . . . . . . . 3135.15 Integrating over the Defective Element . . . . . . . . . . . . . . . . . . . 3165.16 Mohr or the Weak Form of Influence Functions . . . . . . . . . . . . 3195.17 Near and Far . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3215.18 Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3235.19 Integral Bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3305.20 Retrofitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3335.21 Classical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3335.22 Calculation of uc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

5.22.1 Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3395.22.2 Direct Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3415.22.3 Support Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

5.23 Sherman-Morrison-Woodbury . . . . . . . . . . . . . . . . . . . . . . . . . 3455.24 One-Click Reanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

5.24.1 When the Load “Is Hit” . . . . . . . . . . . . . . . . . . . . . . . 3475.24.2 Singular Stiffness Matrices . . . . . . . . . . . . . . . . . . . . . 349

5.25 Subsequent Installation of Joints . . . . . . . . . . . . . . . . . . . . . . . 3495.26 Buckling Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

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5.27 Dynamic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3515.27.1 Antisymmetry in the Compensating Motions . . . . . . . . 351

5.28 The Continuum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3525.28.1 Potential Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3525.28.2 Kernels jþ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3565.28.3 The Two Approaches . . . . . . . . . . . . . . . . . . . . . . . . . 358

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

6 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3616.1 Singular Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3616.2 Singular Support Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3626.3 Single Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3636.4 Decay of Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3676.5 Infinite Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3696.6 Symmetry of Adjoint Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 3726.7 Cantilever Wall Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3746.8 Standard Situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3786.9 Singularities in Influence Functions . . . . . . . . . . . . . . . . . . . . . 378

7 Mixed Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3877.1 Bernoulli Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3877.2 Timoshenko Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3887.3 Poisson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3897.4 The Plate Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3907.5 Kirchhoff Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3927.6 Reissner-Mindlin Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3947.7 Influence Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3967.8 Betti Extended . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

8 Nonlinear Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4018.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4018.2 Gateaux Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4028.3 Nonlinear Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

8.3.1 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4048.4 Geometrically Nonlinear Beam . . . . . . . . . . . . . . . . . . . . . . . . 404

8.4.1 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . 4068.5 Geometrically Nonlinear Kirchhoff Plate . . . . . . . . . . . . . . . . . 4088.6 Nonlinear Elasticity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

8.6.1 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4108.6.2 A Truss Element in 3-D . . . . . . . . . . . . . . . . . . . . . . . 4118.6.3 Planar Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

8.7 Nonlinear Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

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9 Addenda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4219.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4219.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4239.3 FE-Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4249.4 Vectors and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4259.5 The Algebra of the Identities . . . . . . . . . . . . . . . . . . . . . . . . . . 4269.6 The Algebra of Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . 4299.7 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . 4319.8 Vh and V þ

h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4339.9 Galerkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4349.10 Weak Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4359.11 Variation and Green’s First Identity . . . . . . . . . . . . . . . . . . . . . 4389.12 The Basic Functional (Hu-Washizu) . . . . . . . . . . . . . . . . . . . . . 4399.13 Force Method and Slope Deflection Method . . . . . . . . . . . . . . . 4409.14 The Adjoint Operator and Green’s Function . . . . . . . . . . . . . . . 4409.15 Rope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4439.16 Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4439.17 Poisson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4459.18 Potentials and Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4469.19 Single Force Acting on a Plate . . . . . . . . . . . . . . . . . . . . . . . . 4469.20 Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4489.21 The Dimension of the fi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4499.22 Weak and Strong Influence Functions . . . . . . . . . . . . . . . . . . . 4509.23 How the Embedding Theorem Got Its Name . . . . . . . . . . . . . . 4519.24 Point Loads and Their Energy . . . . . . . . . . . . . . . . . . . . . . . . . 4559.25 Early Birds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

10 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

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